A couple has two children, one of whom is a girl. If the probability

Question

A couple has two children, one of whom is a girl. If the probability of having a girl or a boy is 50%, what is the probability that the couple has two daughters?

A. 1/8
B. 1/4
C. 1/3
D. 1/2
E. 2/3

eshan429 · Accepted Answer

If the probability of having a boy or a girl is  50% , the possibilities can be  (BB, BG, GB, GG)

Now, it's given that one of the kids is a girl. So, the sample set reduces to  (BG, GB, GG)

Probability of having two girls =  \frac{1}{3}

Maxirosario2012 · Answer

Hi...
I don´t understand why permutation matters here. I think this problem is about combinatorics. 
In the family there is already one girl. Then the probability of a second girl is 1/2. 
Why does it matter whether the girl is the first or the second in the family?. I know that I am wrong but I don´t understand why.

Anamika2014 · Answer

The question never says that the first child is a girl. If it was so, then the probability would indeed be  \frac{1}{2} .

But here we need to consider following two cases:

1) The first child is a girl 
In this case there are two possibilities -  GB, GG .

1) The first child is a boy 
In this case there is just one possibility -  BG .
Notice that  BB  is not possible as the question explicitly states that one girl is a child.

Hence the probability is  \frac{1}{3} .

eltonj · Answer

I still do not understand this... How is GB different than BG?

If you walk up to a couple and already know that A) they have two children and B) one is a girl....

then half the time they will have two girls and half the time they will have a boy and a girl...

Can someone confirm this? Seems like people are regurgitating information straight from the solutions instead of actually thinking through the problem.

Bunuel · Answer

First child ---- Second child: 
B ---------------------------- B;
 B ---------------------------- G;
G ---------------------------- B;
G ---------------------------- G.

We know that one if the kids is a girl. So, we have one of the three cases in red. Each case there is equally likely (each has the probability of 1/2*1/2=1/4). So, the probability of GG is 1/3.

Answer: C.

Hope it's clear.

eltonj · Answer

Hello, thank you kindly for your response. I fully understand how an answer of 1/3 is found. That being said, I am still not sure why BG and GB is being treated as two separate cases. In my mind, the problem looks like this:

Child Child 
B ----------- B
 B ----------- G
G ----------- G

I don't see the need to have BG and GB. This problem has nothing to do with the order of the children. We know they have a Girl, so they either have a boy and a girl or two girls.

Bunuel · Answer

Having an older girl and a younger boy is different from having an older boy and a younger girl.

eltonj · Answer

Age is 100% irrelevant in this question, all we care about is gender. The answer is 1/2.

Bunuel · Answer

The correct answer is 1/3, not 1/2. Having a boy and a girl is twice as likely than having two boys (or two girls), 1/2 and 1/4 respectively, so you should take this into account.

eltonj · Answer

Ahhh okay this makes much more sense to me. Thank you very much!

jumper41 · Answer

I was having the same problem as eltonj - but after reading through this thread, Bunuel 's last explanation cleared it up. Thank's Bunuel . Here's an interesting link that shows an interesting visual explaining conditional probability: https://setosa.io/conditional/

TimeTraveller · Answer

Sample space = {bb,bg,gb,gg}

Favourable event = {gg}.

Since it's given that one of the child is a girl, the new Sample Space = {bg,gb,gg}.

So, probability = 1/3. Ans (C).

chetan2u · Answer

Hi,
we know one is girl..

the cases possible with one kid as girl is
BG or GB or GG -- 3 ways
we are talking of prob of GG -- 1 way
ans  \frac{1}{3} 
C

REDWASP2205 · Answer

chetan2u

But knowing One of them is a girl, shouldn't we remove BG from the possible cases.

chetan2u · Answer

No, Because we just know that one of them is G..
here the order is important but we are not given the order but just that one is a G..
If it said the first one is a G, then yes we would have had ONLY two cases GB or GG..

Merged topics

HKD1710 · Answer

Hi  chetan2u ,

This one is crucial for me to understand. whether a boy is first and girl is second or boy is younger than girl or vice versa--WHY do we consider BG, GB as two cases instead of ONE case only. We are not arranging here.

I looked at the responses from Bunuel and others on original post too but i still have this doubt. Could you please explain this?

chetan2u · Answer

Hi,

I would relate this Q to two different Q.

1) Ways to pick two books out of 4 books randomly?
4C2..

2) Ways to pick two books out of 4 books one after another?
this will be 4C2 *2!

Our Q is similar to the 2nd case

Abhishek009 · Answer

Hello HKD1710 Please go Through eshan429 's post probability-217147.html#p1406381 IMHO, I don't feel you need to consider younger /older in this case. The chances of having a twins ( Boy and Girl) in a single delivery can not be rejected. Both Boy and Girl may be born at the same time, we need to consider only the probability of having only 2 daughters ( They may be twins as well - Born on the same day) Younger / Elder daughter is not important in this case , we simply restrict ourself to a female child (daughter) Happy Preparations Abhishek

HKD1710 · Answer

Thanks  Abhishek009 ,

I went thru all those solutions and i still did not understand only the following:

A couple has two children, one of whom is a girl. If the probability

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A couple has two children, one of whom is a girl. If the probability

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